For integer $m>0$ and $0<q<m/2$, consider the sum $$ S(m,q) = \sum_{i=0}^{m-q} \binom{m}{i} \binom{m-i}{q}^2. $$ Is there any standard technique to find the asymptotic for this sum, or at least to determine its order of magnitude, as $m$ grows? (In case it matters, $q=cm$ with fixed $c<1/2$ can be assumed.) Thanks!